Choosing Intellectual Protection: Imitation, Patent Strength and Licensing∗ David Encaoua†and Yassine Lefouili‡ Revised Version: March 2006

Abstract This paper investigates the choice of intellectual protection for a process innovation. We set up a multi-stage model in which the decision whether or not to patent an innovation is affected by three parameters: the patent strength defined as the probability that the right is upheld by the court, the cost of imitating a patented innovation relative to the cost of imitating a secret innovation, and the innovation size defined as the extent of the cost reduction. We find that large innovations are likely to be kept secret whereas small innovations are always patented. Furthermore, medium innovations are patented only when patent strength is sufficiently high. Finally, we investigate a class of licensing agreements that may be used to settle patent disputes between patent holders and their competitors.

Keywords: Patent, Secrecy, Imitation, Licensing. JEL classification: D45, L10, O32, O34.

∗

We thank two anonymous referees for helpful comments and suggestions. University of Paris-I Panth´eon Sorbonne, Centre d’Economie de la Sorbonne, 106-112, Bd de l’Hˆ opital 75647 cedex 13, Paris, France. E-mail: [email protected] ‡ University of Paris-I Panth´eon Sorbonne, Centre d’Economie de la Sorbonne, 106-112, Bd de l’Hˆ opital 75647 cedex 13, Paris, France. E-mail: [email protected] †

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1

Introduction

The traditional view that firms always prefer patents to other forms of protection for their innovations has been empirically challenged for a long time. It is now well known that secrecy, first mover advantage and exploitation of lead time may be preferred forms of protection in many industries. Even in the same industry, forms of protection may differ according to the nature and importance of the innovation and to the disclosure effect. Early studies by Scherer (1965, 1967, 1983) have shown that the propensity to patent varies significantly accross industries and that inter-industrial variations in patenting activity are not explained by R&D expenditures. Pakes and Griliches (1980) were among the first to find that the degree of randomness in the patenting activity within industries was not explained by R&D variations. They have shown that the residual patenting behaviour was explained by the potential imitation allowed by the disclosed information and by the innovator’s capability to appropriate the rents generated by the innovation. Mansfield (1986) obtained similar results based on a survey where US manufacturing firms were asked what fractions of inventions they would not have developed in the absence of patents between 1980 and 1983. These fractions were very low in many industries (less than 10% in electrical equipment, primary metals, instruments, motor vehicles and others) and relatively high in industries like pharmaceuticals (60%) and chemicals (40%). Two more recent surveys (Yale Survey by Levin et al., 1987 and Carnegie Mellon Survey by Cohen et al. 2000) confirm these trends: it is only in industries where knowledge is strongly codified that patents appear to be substantially preferred to other forms of protection. Despite this accumulated empirical evidence, theoretical explanations of why and when an innovator would prefer to keep an innovation secret rather than patent it remain rather scarce. Before turning to the related literature, note that even if patenting is not considered as the best form of protection, innovators have a lot of reasons to apply for patents serving purposes different from protection (Hall and Ziedonis, 2001, Encaoua et al. 2004). This feature complicates the problem. Indeed, a theoretical explanation of why and when patenting is not the best form of protection must also be consistent with the more cumbersome issue of why, despite the existence of preferred forms of protection, patents remain so widespread (Scotchmer, 2004). At least three types of theoretical arguments are required to explain the protection choice. First, patents must be recognized as not being ironclad property rights but rather probabilistic rights. Lemley and Shapiro (2005) qualify the uncertain nature of patents in a suggestive way: ”A patent does not confer upon its owner the right to exclude a potential imitator but rather a right to try to exclude by asserting the patent in court. When a patent holder asserts its patent against an alleged infringer, the patent holder is rolling the dice. If the patent is found invalid, the property right will have evaporated ”. Thus, patent strength refers

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to the probability to recover damages, with the consequence that only strong patents give in principle to the patent holder the right to exclude an infringer or to force him to buy a license. But as we argue below, even holders of weak patents may escape the uncertain litigation process. They succeed through their licensing strategies to capture a significant part of the consumers’ surplus. This is why the notion of patent quality enters so forcefully in the agenda of antitrust authorities nowadays, especially through the criticism of the examination system in the Patent Office (Merges, 1999, Lemley, 2001). Second, the traditional view that knowledge is a blueprint has also been challenged. Replicating an existing invention may be costly and time consuming because knowledge may be more embedded in individuals and firms rather than physical products or equipment. One has to distinguish innovations according to their secrecy effectiveness, which is the main determinant of the imitation cost (Anton et al. 2005). Many innovations involve hidden know-how even if the allowed performance is perfectly observable. Consider for instance a process innovation leading to a cost reduction that is reflected back in the market price but for which the technological knowledge is neither perfectly revealed nor easily reverse-engineered. In this case, imitating the process innovation or building around it may be rather costly. Moreover, the imitation cost may depend on whether the invention has been patented or not. As the patent discloses some enabling technological information, it is clear that imitating a patented innovation should be at most as costly as if it was kept secret. Our paper offers a natural framework to analyze the classical trade-off between getting a legal protection involving a compulsory disclosure of enabling information and keeping secrecy by giving up legal protection. Third, even if patents do not always appear as the best form of protection, innovators may nevertheless prefer to patent their innovations because holding a patent offers the possibility to settle a dispute against an alleged infringer through a licensing agreement (Lemley and Shapiro, 2005, Farrell and Shapiro, 2005). Alleged infringers may also prefer to avoid a litigation process not only because litigation is costly but also because winning the lawsuit against the patent holder involves a free-riding aspect, as other competitors benefit from the asserted patent’s invalidity. Therefore, even when they are weak, patents generate substantial revenues through licensing royalties that may harm consumers. This is why patent settlements raise serious concerns for competition policy authorities (Shapiro, 2003, Encaoua and Hollander, 2004). The main objective of this paper is to introduce these arguments in a simple model allowing a discussion of the following issues: i/ What are the different forces that interact in the choice of a protection regime (patent or secrecy)? ii/ How are these forces affected by the patent strength, imitation cost and innovation size? iii/ What licensing agreements are likely to be used to settle a patent dispute between a patent 3

holder and an alleged infringer? Two main contributions have explicitly explored the decision whether to patent an innovation or not.1 Hortsman et al. (1985) assume that an innovating firm possesses private information about profits available to competitors and that patent coverage may not exclude profitable imitation. Conceived as an information transfer mechanism, a patent that covers full information is not optimal. The optimal innovator’s choice is a mixed strategy between patenting and keeping secrecy while the follower’s optimal choice is to stay out of the market when the innovator patents and to imitate when the innovator does not. The peculiarities of this model, in terms of the patent’s signalling aspect and the a priori restrictions put on the follower’s action, explain why imitation of a patented innovation does not occur at equilibrium. Since our paper is close to Anton and Yao (2004), we describe more thoroughly their framework. Starting from the premise that disclosure provides competitors with usable information and focusing on the innovator’s decision about how much of an innovation should be disclosed, their model is particularly relevant for a special type of secrecy effectiveness. They describe a situation where the real innovation performance is not directly observable while the disclosed know-how enables a competitor to costlessly replicate it. Therefore, by choosing the amount to be disclosed, the innovator directly controls the behaviour of the potential imitator. Their model is a signalling game where the innovator has private information on the innovation size and decides to reveal partially or fully this information, letting the potential imitator infer the leader’s advance. The follower chooses either to imitate or not under the risk of infringement. A refined perfect bayesian equilibrium of the signalling game involves a separating strategy in which: i/ small innovations are patented and fully disclosed; ii/ medium innovations are patented and partially disclosed; iii/ large innovations are not patented and partially disclosed through a public announcement. This result is illustrated by their suggestive title: ”little patents and big secrets”. In our model, we maintain the general tradeoff to which an innovator is confronted when choosing the protection regime. However, rather than focusing on the signalling aspect we assume that: i/ the process innovation size, measured by the cost reduction, is directly observable; ii/ a patent reveals technological information that lowers the imitation cost relative to the situation where the know-how embedded in the innovation is kept secret.2 Choosing 1 Among other papers related to the choice of an intellectual property regime, one can mention Crampes (1986), Gallini (1992) and Scotchmer and Greene (1990). Crampes (1986) examines the tradeoff between keeping secret an invention during an indefinite time or obtaining a legal protection over a finite duration (the statutory patent life). Gallini (1992) introduces the idea that breadth governs the cost of inventing around the patent. However, it is entry cost rather than imitation cost that matters. Scotchmer and Greene (1990) focuses on the impact of patent policy on the incentives to innovate. Their model involves a binary choice as the innovation would not be realized if it were not patented. They also assume full disclosure of technological know-how. 2 If the reduction of the imitation cost directly depends on the disclosed level of enabling knowledge, using a relative imitation cost parameter is equivalent to using a disclosure level. It appears however that working with the relative imitation cost parameter is more convenient since the extent of imitation remains controlled

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to patent may expose the innovator either to an increased imitation level or to a lower one because the imitation level does not only depend on the imitation cost but also on two other crucial parameters: the innovation size and patent strength. It may happen that an innovator benefits from being imitated : this occurs whenever the incurred loss due to imitation is overcompensated by the damages it receives from an imitator if the court upholds the patent validity and the patent infringement. If patent protection and secrecy lead to the same imitation level, then patenting is preferred since damages are expected under the patent regime. This corresponds to the damage effect. But whenever imitation levels differ according to the protection regime, a conflict arises as long as imitation becomes higher under the patent regime. This corresponds to the competition effect. Therefore, when the imitation extent is decided by the follower, different interactions may occur between the competition effect and the damage effect.

Our paper aims to clarify these interactions. We propose a complete

information multistage game in which three common knowledge parameters are important: the innovation size, the patent strength and the relative cost of imitation. We depart from the assumption limiting a priori an imitator’s behaviour by letting it choose its own cost reduction in response to the process innovation. We also assume that the damages paid by an imitator infringing a valid patent are equal to its profit, which corresponds to the unjust enrichment legal doctrine. Our damages specification only presents a slight difference with the infringer’s revenue rule assumed in Anton and Yao (2004).3 Thus, our results may be compared to the ”little patents and big secrets” results in Anton and Yao. Our main findings are as follows. For a given innovation size, patent strength and relative imitation cost generally act as substitutes: A decrease in one of these parameters must be compensated by an increase in the other one in order to keep the same value of the innovator’s profit. Inventors of small process innovations always prefer patent protection to secrecy. This reminder of the ”little patents” result by Anton and Yao (2004) rests however on a different argument in our model. For large process innovations, our results present some difference with the ” big secrets” characterization in Anton and Yao. Our model does not totally discard the possibility of patenting some large process innovations, whenever imitation is too costly. This may happen when information is poorly disclosed in the patent. In this case the innovator is indifferent between secrecy and patent protection. For medium process innovations, our results differ more significantly from those of Anton and Yao. It is not optimal for a firm producing such an innovation to file a patent of bad quality, that is a patent having a low probability to be upheld by the court, unless the disclosed information does not significantly by the imitator, while the choice of a protection regime is made by the innovator. 3 In a more recent paper, Anton and Yao (2005) introduce the ”lost” profits of the patentee, defined as the profits that would have occured in the absence of infringement. They show that at equilibrium, infringement may take one of two forms: a ”passive” form in which lost profits of the patentee are zero and an ”agressive” form where they are positive. One of the main results in Anton and Yao (2005) is that infringement always occurs when damages equal lost profits. This last result does not hold in our model.

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lower the imitation cost. We show that there exists a safe protection level that is sufficient to deter any imitation and that this level is lower than a 100% protection. As the innovation size decreases, protecting the innovation by means of secrecy becomes less likely. Finally, the ”one size fits all” principle in the patent design is not validated by our analysis. These results raise many practical issues. While the model predicts that it is seldom optimal for a firm to file a patent when the probability that it will be upheld by the court is low, bad quality patents (relatively to novelty and non-obviousness requirements) are widespread in real word. How can this be explained? Moreover, why are low quality patents not litigated more often than we observe? These issues are at the heart of what has been called the patent paradox (Hall and Ziedonis, 2001, Scotchmer, 2004). We devote a brief analysis that suggests a possible answer to explain this paradox. Whenever a patent is not conceived only as a protection against imitation but also as a tool to reach private settlements through licensing agreements (Shapiro, 2003), licensing agreements may act as an alternative to patent litigation. A royalty rate independent of the patent strength combined to a specific fixed fee may serve this purpose. The remainder of the paper is organized as follows. The model is presented in section 2. The market competition outcome under the shadow of infringement is described in section 3. The imitator’s behavior is analyzed in section 4. The protection regime choice is examined in section 5. We devote section 6 to licensing agreements. Our conclusions are presented in section 7.

2

The basic set-up

We examine a process innovation in a framework involving two competing firms. We suppose that firm 1 is an innovating firm and firm 2 is a potential imitator. Each firm is risk-neutral and seeks to maximize its expected profit. Initially, both firms produce at the same marginal cost c > 0 . Fixed production costs are assumed to be zero. We assume that firm 1 undertakes an R&D investment which reduces its marginal cost to the level d1 < c. The game we consider below starts once the innovation is realized and involves three stages. First, in the protection stage, the innovator has to choose between two protection regimes. The first regime, which we denote by P, is to patent its innovation, and the second regime, which we denote by S, consists of protecting its innovation by means of secrecy. Second, in the imitation stage, after the observation of the innovator’s marginal cost d1 , firm 2 chooses whether, and the extent to which, it wishes to imitate (or ”build around”) the innovator’s technology. It makes an investment to reduce its marginal cost to d2 ∈ [d1 , c] . Note that we do not allow the imitator to improve the innovator’s technology.4 The difference 4

Green and Scotchmer (1995) and Chang (1995) examine the issue of technology improvement by a second generation innovator in a cumulative innovation framework.

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c − d2 captures the ”extent of imitation”. When d2 = c, there is no imitation at all and when d2 = d1 , imitation is full. We assume that imitation at a level d2 ∈ [d1 , c[ induces a fixed imitation cost I(d2 ) which depends on whether the innovation is patented or kept secret. Precisely, we assume that the imitation cost under the patent regime, which we denote by I P (d2 ), and the imitation cost under the secrecy regime, which we denote by I S (d2 ), satisfy the following condition : I P (d2 ) = f I S (d2 ) where the parameter f ≥ 0 measures the relative costs of imitation under the regimes P and S. We assume that f ≤ 1 : since patenting involves a compulsory disclosure, it is likely that imitating a patented innovation will turn out to be less costly than imitating a secret innovation. Third, in the competition stage, market outcomes are determined under the shadow of punishment. We assume that, when the innovation is patented, firm 1 sues firm 2 for infringement if firm 2 imitates firm 1 by adopting a follow-up technology that reduces its marginal cost to d2 < c. We also assume that firm 2 systematically contests the validity of the patent covering the innovation.5 We denote by g the probability that the patent survives the imitator’s legal contestation of the patent’s validity. We interpret this parameter as the patent’s quality: low quality patents have higher chances to be invalidated by a court than high quality patents. Thus, a higher quality patent (in terms of novelty and inventiveness) is less uncertain: the probability that a court will uphold its validity is higher. We denote by e the probability that an imitation infringes the innovator’s patent conditional on the patent being valid. This probability can be interpreted as an indicator of the lagging patent’s breadth: the broader the patent’s breadth, the higher the probability that a follow-up technology that reduces the marginal cost c to d2 ∈ [d1 , c[ is an infringement of the patent on the process innovation d1 . Firm 2 is compelled to pay damages, supposed to be equal to its market profit, if and only if the patent is held valid and the imitation infringes the patent. This occurs with probability θ = eg. The parameter θ ∈ [0, 1] , which we assume to be common knowledge,6 corresponds to what it is called the patent strength (Shapiro 2003). When the innovation is not patented, no damages are paid. Following Anton and Yao (2004), we model our duopoly market competition as a traditional Cournot competition with linear market demand: p(x1 + x2 ) = a − (x1 + x2 ) 5

The typical defense in real-world patent infringement suits is to contest infringement and patent validity. Assuming that the parameter θ is common knowledge is consistent with our goal of putting aside any signalling effect. However, this assumption might seem to be quite strong: It is reasonable to think that the innovator might have some private information as to the validity of its patent (captured through the parameter g) and the imitator might have some private information as to the probability of infringement e. See Bebchuk (1984) and Meurer (1989) for analysis of patent litigation under asymmetric information. 6

7

where x1 is the output of firm 1, x2 is the output of firm 2 and p(x1 + x2 ) is the market clearing price. We assume that c < a < 2c. The first inequality is usual and means that the marginal cost before innovation is below the choke price. The second inequality expresses that the market is small which is a likely scenario for innovative markets, as it allows the possibility that the innovative firm becomes at least twice as efficient as it currently is. In other words, the inequality a < 2c implies that there exist innovations d1 such that d1 < 2c − a, which can also be written as a − d1 > 2(a − c). We choose a convex specification for the imitation technology and, to reach analytical results, we use the following quadratic expression : ( I(d2 ) =

3

f (c−d2 2 )

2

(c−d2 )2 2

if

P atent

if

Secret

Competition stage

Competition occurs under the shadow of litigation only if the innovation is patented. Therefore, the outcome of the competition stage depends on whether the innovation is patented or not.

3.1

Patented innovation

We separately examine the cases d2 < c (the follower imitates the innovator, at least partially) and d2 = c (the follower does not imitate the innovator), as the profit functions differ between these two cases. The follower imitates (d2 < c) : Under regime P , the expected gross profits of firm 1 and firm 2 are given by: ΠP1 (x1 , x2 , d1 , d2 , θ) = (a − (x1 + x2 ) − d1 ) x1 + θ (a − (x1 + x2 ) − d2 ) x2 | {z } | {z } market profits

expected damages

and ΠP2 (x1 , x2 , d2 , θ) = (1 − θ) (a − (x1 + x2 ) − d2 ) x2 From the expected profits, one derives the Cournot-Nash equilibrium outputs xP1 (d1 , d2 , θ) and xP2 (d1 , d2 , θ). They correspond either to an interior solution where both firms are active: xP1 (d1 , d2 , θ) ≥ xP2 (d1 , d2 , θ) > 0 or to a boundary solution where only firm 1 is active : xP1 (d1 , d2 , θ) > xP2 (d1 , d2 , θ) = 0.

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Consider first an interior solution. Routine computations lead to: xP1 =

a(1 − θ) + d2 (1 + θ) − 2d1 3−θ xP2 =

a − 2d2 + d1 3−θ

A necessary and sufficient condition for an interior solution to exist is : d2 <

a + d1 2

(1)

Note that this condition is always satisfied when d1 > 2c − a. Note also that the market price pP (θ) is given by pP (θ) =

a+d2 (1−θ)+d1 3−θ

which is increasing in θ ∈ [0, 1] as long as condition

(1) is satisfied. Consider now a boundary solution. Such a solution arises when condition (1) is not satisfied and is characterized by: xP1 =

a − d1 2

and xP2 = 0

The follower does not imitate (d2 = c) : The equilibrium outputs in this case can be derived from those of the previous case by taking θ = 0 and d2 = c. Hence: - If d1 > 2c − a, then the equilibrium outputs are given by: xP1 =

a + c − 2d1 3

and xP2 =

a + d1 − 2c 3

- If d1 ≤ 2c − a , then we have the same boundary solution as in the imitation case: xP1 =

a − d1 2

and xP2 = 0

Summing up these cases, the expected equilibrium gross profits depend on d1 , d2 and θ in the following way : • If d1 ≤ 2c − a then:   [a−d1 (2−θ)+d2 (1−θ)][a(1−θ)−2d1 +d2 (1+θ)]+θ[a−2d2 +d1 ]2 (3−θ)2 ΠP1 (d1 , d2 , θ) = 2 (a−d 1)  4

( ΠP2 (d1 , d2 , θ)

=

2

2 +d1 ) (1 − θ) (a−2d (3−θ)2

0 9

if

d2 <

if

d2 ≥

a+d1 2 a+d1 2

if if

d2 < a+d1 2

a+d1 2

≤ d2 ≤ c (2) (3)

• If d1 > 2c − a then:   ΠP1 (d1 , d2 , θ) = 

[a−d1 (2−θ)+d2 (1−θ)][a(1−θ)−2d1 +d2 (1+θ)]+θ[a−2d2 +d1 ]2 (3−θ)2 (a+c−2d1 )2 9

( ΠP2 (d1 , d2 , θ)

=

2

2 +d1 ) (1 − θ) (a−2d (3−θ)2

(a+d1 −2c)2 9

if

d2 < c

if

d2 = c

if

d2 < c

if

d2 = c

(4)

Therefore, under Cournot competition, firm 2 is driven out of the market if it keeps its old technology when the innovation is large enough (d1 < 2c − a) and remains active on the market (even without imitating firm 1) when the cost reduction innovation is small enough (d1 > 2c − a). This result depends on the small market assumption (a < 2c). In a large market (a > 2c), firm 2 would remain in the market whatever the innovation size. Thus, the small market assumption allows for strategic aspects that would not be captured otherwise.

3.2

Non-patented innovation

The equilibrium outcomes under the secrecy regime are derived from those under the patent regime by taking θ = 0. This simply means that no damages are paid when imitation occurs under secrecy. One obtains : ( ΠS1 (d1 , d2 )

= (

ΠS2 (d1 , d2 )

4

=

(a−2d1 +d2 )2 9 (a−d1 )2 4

if

d2 <

if

d2 ≥

(a−2d2 +d1 )2 9

if

d2 <

0

if

d2 ≥

a+d1 2 a+d1 2

(5)

a+d1 2 a+d1 2

(6)

Imitation stage

Firm 2 aims to maximize its net profit when it chooses its imitation level d2 ∈ [d1 , c] . Since the follower’s gross profit and imitation cost depend on whether the innovation is patented or not, we need to distinguish between these two regimes.

4.1

Patented innovation

Under this regime, the imitator’s net profit when it adopts a technology allowing to produce at marginal cost d2 ∈ [d1 , c] is given by : 1 GP2 (d1 , d2 , f, θ) = ΠP2 (d1 , d2 , θ) − f (c − d2 )2 2

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The follower’s optimal imitation level when the innovation is patented is determined as: dP2 (d1 , f, θ) = Arg maxGP2 (d1 , d2 , f, θ) d2 ∈[d1 ,c]

Define A(θ) =

1−θ . (3−θ)2

It is a decreasing function of θ ∈ [0, 1] such that A(0) =

1 9

and A(1) = 0.

The function d2 → H(d1 , d2 , f, θ) = A(θ)(a − 2d2 + d1 )2 − 21 f (c − d2 )2 , which is the expression of GP2 (d1 , d2 , f, θ) when d2 <

a+d1 2

and d2 < c, is necessarily either convex or concave over its

whole definition domain. The following preliminary results are easy to show: 1- The function d2 → H(d1 , d2 , f, θ) is stricly convex if f < 8A(θ) and is strictly concave if f > 8A(θ). 2- The unconstrained extremum of d2 → H(d1 , d2 , f, θ) is easily obtained by the the FOC: dint 2 (d1 , f, θ) = c +

4A(θ)(d1 − 2c + a) 8A(θ) − f

(7)

In order to obtain the value of dP2 (d1 , f, θ), it is necessary to know whether H(d1 , d2 , f, θ) is convex or concave and to compare dint 2 (d1 , f, θ) to d1 and c. For instance, when d1 < 2c − a and f > 8A(θ) (H strictly concave), equation (8) leads to dint 2 (d1 , f, θ) > c. Moreover, as GP2 (d1 , d2 , f, θ) is a discontinuous function in d2 for d2 = c, it is necessary to compare the value of GP2 (d1 , c, f, θ) obtained in the absence of imitation to the value of Arg maxGP2 (d1 , d2 , f, θ) d2 ∈[d1 ,c[

obtained with imitation. 4.1.1

Large innovations (d1 < 2c − a)

With large innovations, partial imitation never occurs. The following proposition (see appendix A1 for the proof) distinguishes two areas in the (θ, f ) space according to whether the optimal imitation level is maximum (i.e. dP2 (d1 , f, θ) = d1 ) or minimum (i.e. dP2 (d1 , f, θ) = c). In the (θ, f ) space, the extent of each of these two areas depends on d1 . Proposition 1 For large innovations (d1 < 2c−a), there exists a threshold function ρ (d1 , θ) = 2  1 which is decreasing in the patent strength θ and the innovation size c − d1 such 2A(θ) a−d c−d1 that : - If f < ρ (d1 , θ) , then the follower fully imitates: dP2 (d1 , f, θ) = d1 . - If f > ρ (d1 , θ) , then the follower does not imitate dP2 (d1 , f, θ) = c and is driven out of the market.

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The interpretation of this proposition, illustrated in figure 1, is quite clear. When the process innovation is large enough (d1 < 2c − a), there exists a threshold imitation cost ρ (d1 , θ) such that if f is below this threshold it is optimal for a follower to fullly imitate the patented innovation (dP2 (d1 , f, θ) = d1 ), whereas if f is above the threshold, it does not pay  2 1 to imitate. Note that ρ (d1 , 0) = 29 a−d < 89 for any d1 < 2c − a. Therefore, sufficiently c−d1 large patented innovations, even when protected by a weak patent (θ not far from 0), will not be imitated as long as the imitation cost parameter f is sufficiently high (f > 89 ). This result means that for a sufficiently high lead advance of the innovator and a sufficiently high imitation cost, imitation of the patented innovation never occurs and the technological follower is driven out of the market. This result, which occurs under a low intensity of competition in the product market (Cournot competition) is likely to hold under a higher intensity of competition. Another important result is that the threshold imitation cost ρ (d1 , θ) decreases as the patent is stronger (higher θ) and as the innovation is larger (lower d1 ). Hence the size of the region where no imitation occurs increases in the innovation size. Furthermore, 12

the patent strength θ and the imitation cost parameter f act as substitutes, because both f and θ include a cost dimension for the imitator, directly via f and indirectly via θ. As θ increases, the expected damages paid by the infringer increase and correspond to a higher cost of infringement. Therefore an increase in one of these cost parameters must be compensated by a decrease in the other one in order for the imitator to keep the same expected profits. Remark: The fact that partial imitation does not occur in this case rests on two key properties of the Cournot duopoly equilibrium profit under a linear demand (and possibly other demand specifications). These are convexity in the firm’s own marginal cost and submodularity with respect to own marginal cost and rival’s marginal cost. These properties along with the convexity of the imitation cost makes the region of the (θ, f ) space where the imitator’s net profit GP2 (d1 , d2 , f, θ) is convex expands as d1 decreases. In this region, the only potential optimal behaviors for the imitator is to imitate fully or not imitate at all. If d1 is sufficiently low (i.e. the innovation is sufficiently large), this region is sufficiently wide to encompass (high) values of f (for a fixed θ) for which non-imitation is optimal. Since non-imitation will still be optimal for higher values of f , partial imitation will never occur. 4.1.2

Small and medium innovations (d1 > 2c − a)

This case is more complicated to analyze. Partial imitation is no more discarded. Indeed, three situations, namely full imitation, partial imitation and no imitation, may occur according to the values of the parameters (d1 , f, θ). The following proposition (proven in appendix A2) summarizes the follower’s optimal strategy according to the values of the parameters f , θ and d1 when d1 > 2c − a. Proposition 2 Consider small and medium innovations (d1 > 2c − a). For each value of 1 d1 , there exist three separating functions in the (θ, f ) space defined by α (d1 , θ) = 4A(θ) a−d c−d1 , 2  2  8A(θ) −2c+a 1 β (d1 , θ) = 2A(θ) a−d − 29 d1c−d and γ(θ) = 1−9A(θ) that delineate three regions: c−d1 1

- If (f, θ, d1 ) satisfy f < M in(α (d1 , θ) , β (d1 , θ)), then the follower fully imitates: dP2 (f, θ, d1 ) = d1 - If (f, θ, d1 ) satisfy α (d1 , θ) < f < γ(θ), then the follower partially imitates: dP2 (d1 , f, θ) = dint 2 (d1 , f, θ) - If (f, θ, d1 ) satisfy β (d1 , θ) < f < α (d1 , θ) or f > M ax(α (d1 , θ) , γ(θ)), then the follower does not imitate: dP2 (f, θ, d1 ) = c. The functions α (d1 , θ) and β (d1 , θ) are decreasing in the patent strength θ and in the innovation size c − d1 and γ(θ) is decreasing in the patent strength θ. Moreover the equations in θ given by α (d1 , θ) = β (d1 , θ) and α (d1 , θ) = γ(θ) have the same solution θ0 (d1 ) ∈ [0, 1[ which means that the curves f = α (d1 , θ), f = β (d1 , θ) and f = γ(θ) meet at a same point θ0 (d1 ) in the (θ, f ) space for a given d1 > 2c − a.

13

This proposition is illustrated in figure 2 in which we assume that 2c − a < d1 <

9c−4a 5 .

4 a−d1 9 c−d1

that α(d1 , 0) < 1 ⇔ d1 < 9c−4a 5 . Therefore it is worth distin
9c−4a guishing medium innovations 2c − a < d1 < 5 from small innovations (d1 > 2c − a) .

It follows from α(d1 , 0) =

The imitator’s choice of d2 is affected by three variables: the cost parameter f, the patent strength θ and the innovation size c − d1 . Define θγ and θβ (d1 ) respectively as the solutions to the equations: γ(θ) = 1 and β (d1 , θ) = 0. For a given innovation d1 such that d1 > 2c − a, the effect of the cost imitation parameter f on the imitation level d2 depends on the value of the patent strength θ in [0, 1] in a specific way that we now describe. When θ < θγ , the patent is very weak and imitation occurs whatever the cost imitation parameter f for two reasons. First, the risk of infringing a very weak patent is not sufficiently dissuasive: even if an infringement lawsuit occurs, damages will be paid with a very low probability θ. Second, imitation is not expensive enough to deter imitation of a small or medium innovation (d1 > 2c − a). Therefore, imitation is either partial or full according to the imitation cost f . It is only partial if f is above the threshold α (d1 , θ) and it is full if f is below this threshold. When θγ ≤ θ ≤ θ0 (d1 ), the patent is stronger and imitation becomes more expensive since the payment of damages occurs with a higher probability θ. Therefore, imitation may be either absent, partial or full, according to the imitation cost parameter value f . There is no imitation at all when f is higher than γ(θ). Imitation is only partial when f is below γ (θ) and above the previous threshold α (d1 , θ) and is full when f is lower than α (d1 , θ).

14

A third situation occurs when θ0 (d1 ) < θ < θβ (d1 ). In this case, infringing is much more expensive because the patent will be upheld by the court with a higher probability θ. However, keeping the old technology d2 = c is also very detrimental to the follower. Therefore, imitation is either full or absent according to whether f is below or above the lower threshold β (d1 , θ) . Finally, a fourth situation occurs for the highest values of θ (i.e. θ > θβ (d1 )). In this case, it is no longer profitable to imitate even when imitation is costless, because the patent protection is very strong. The imitation cost does not matter anymore and the presumptions that the patent will be upheld by the court and that imitation will be judged as being an infringement are so high that the patent protection plays fully its role against imitation. Since θβ (d1 ) < 1, it is interesting to note that less than perfect protection is sufficient to deter imitation. Therefore, it is justified to refer to the value θβ (d1 ) as the safe protection level. A patent that protects against imitation does not need to be 100% perfect and the safe protection level depends on the importance of the innovation itself. As the innovation is less important (d1 increases), the safe protection level θβ (d1 ) increases. This important result suggests that smaller innovations require stronger protection, since they are likely to be imitated. This is a serious argument against the ”one size fits all” protection principle. The effect of d1 over dP2 (f, θ, d1 ) for a given (θ, d1 ) is interesting: as d1 decreases, leading to an innovation involving a higher cost reduction, the partial imitation area increases because α (d1 , θ) decreases, the full imitation area decreases because both α (d1 , θ) and β (d1 , θ) decrease and the no-imitation area increases because β (d1 , θ) decreases.

4.2

Non-patented innovation

The follower’s optimal imitation strategy under regime S can be simply derived from its optimal imitation strategy under regime P by taking f = 1 and θ = 0. The next proposition summarizes our findings when the innovator chooses to use secrecy to protect its innovation. Proposition 3 Under the secrecy regime, the follower’s optimal imitation strategy d1 → dS2 (d1 ) is given by: - If d1 ≤ 2c − a then the follower does not imitate (dS2 (d1 ) = c) and is driven out of the market. - If 2c − a < d1 < - If

9c−4a 5

9c−4a 5

then the follower partially imitates (dS2 (d1 ) = 9c − 4(a + d1 ) < d1 ).

≤ d1 ≤ c then the follower fully imitates (dS2 (d1 ) = d1 ).

Note that large innovations (d1 ≤ 2c − a ) are never imitated under regime S while they are fully imitated under regime P when f < ρ(d1 , θ). The explanation of this rather unintuitive result simply derives from the previous remark that ρ(d1 , 0) < 1 for any d1 ≤ 2c−a.Then under the patent regime where some enabling knowledge is disclosed, the cost imitation parameter f may be so low (more specifically f < ρ(d1 , θ)) that it may be profitable to incur the low imitation cost f I(d1 ) even when damages are paid with a high probability θ. 15

5

Protection stage

Which protection regime will the innovator choose given its anticipation of the follower’s imitation level under each of the two regimes? To answer this question, we need to compare its equilibrium expected profit under the patent regime ˜ P (d1 , f, θ) ≡ ΠP (d1 , dP (d1 , f, θ), θ) Π 1 1 2 to its equilibrium profit under the secrecy regime ˜ S1 (d1 ) ≡ ΠS1 (d1 , dS2 (d1 )) Π Let us first determine the forces that drive the innovator’s protection regime choice. Consider two (exogenous) imitation levels d2 , d02 ∈ [d1 , c]. The difference ΠP1 (d1 , d2 , θ) − ΠS1 (d1 , d02 ) can be decomposed in the following way:

ΠP1 (d1 , d2 , θ) − ΠS1 (d1 , d02 ) = ΠP1 (d1 , d2 , θ) − ΠS1 (d1 , d2 ) + ΠS1 (d1 , d2 ) − ΠS1 (d1 , d02 ) The first term of this decomposition, namely the difference ΠP1 (d1 , d2 , θ) − ΠS1 (d1 , d2 ), corresponds to what we call the damage effect. Given an imitation level d2 , the innovator can expect some damages if it patents its innovation, which is not the case if it chooses to keep it secret. Let us show that the damage effect is always nonnegative and nondecreasing in θ. This P is equivalent to show  thatfunction θ −→ Π1 (d1 , d2 , θ) is nondecreasing for any d2 ∈ [d1 , c]. 1 , one obtains: For any d2 < min c, a+d 2

6θ(a − d2 )(d2 − d1 ) + 6(a − 2d2 + d1 ) + 2(a − 2d2 + d1 )2 ∂ΠP1 (d1 , d2 , θ) = ∂θ 3−θ   1 When d2 < min c, a+d , one can check that this derivative is strictly positive. In particular, 2 this leads to : ΠP1 (d1 , d2 , θ) > ΠP1 (d1 , d2 , 0) = ΠS1 (d1 , d2 ) When d2 ≥ θ. In particular,

a+d1 P 2 , we have shown that Π1 (d1 , d2 , θ) does not depend on the parameter ΠP1 (d1 , d2 , θ) = ΠP1 (d1 , d2 , 0) = ΠS1 (d1 , d2 ). It follows that ΠP1 (d1 , d2 , θ) −

ΠS1 (d1 , d2 ) ≥ 0 for any θ ∈ [0, 1] and d2 ∈ [d1 , c] . Turn now to the second term. The difference ΠS1 (d1 , d2 ) − ΠS1 (d1 , d02 ) corresponds to what we call the competition effect. The innovator’s profits decline as it is more imitated: ΠS1 (d1 , d2 ) is an increasing function of d2 , which implies that the sign of ΠS1 (d1 , d2 ) − ΠS1 (d1 , d02 ) is the same as the sign of d2 − d02 . Thus, if the innovator anticipates that it will be less (or equally) imitated under the patent

16

regime than under the secrecy regime then both the damage effect and the competition effect suggest the same protection regime, namely the patent regime. But if the innovator anticipates that it will be more imitated under the patent regime than under the secrecy regime, then the damage effect and the competition effect are opposite. The first one increases the incentives for the innovator to choose the patent regime while the second one increases the incentives to choose the secrecy regime. The following lemma, that summarizes and completes what precedes, is useful for the subsequent analysis: Lemma 4 If the innovator is less (or equally) imitated under regime P than under regime S then its optimal protection regime is the patent regime P. In particular, when the innovator anticipates that it will be fully imitated under the secrecy regime or that it will not be imitated at all under the patent regime, it chooses to patent its innovation. In order to determine the innovator’s optimal protection regime, we distinguish the three cases that appeared in the imitation stage discussion. Case 1: d1 < 2c − a (large innovations) In this case, we know that the innovator is not imitated at all when it chooses to keep secrecy (dS2 (d1 ) = c). Its equilibrium expected profit under regime S is then given by: 2 ˜ S (d1 ) = ΠS (d1 , d1 ) = (a − d1 ) Π 1 1 4

We have also shown that under the patent regime, the innovator is fully imitated or not imitated at all, according to whether f < ρ (d1 , θ) or f > ρ (d1 , θ) . Hence, its equilibrium expected profit under regime P is given by: ˜ P (d1 , θ, f ) = Π 1

  

(a−d1 )2 (3−θ)2 (a−d1 )2 4

if

f < ρ (d1 , θ)

if

f > ρ (d1 , θ)

˜ P (d1 , θ, f ) < Π ˜ S (d1 ) if f < ρ (d1 , θ) and Π ˜ P (d1 , θ, f ) = Π ˜ S (d1 ) if f > implying that : Π 1 1 1 1 ρ (d1 , θ) . This leads to the following proposition illustrated in figure 3. Proposition 5 If the innovation is large enough (d1 < 2c − a), the innovator prefers to keep its innovation secret if f < ρ (d1 , θ) and is indifferent between patent protection and secrecy if f > ρ (d1 , θ).

17

Hence, keeping secrecy is always an optimal strategy to the innovator when the innovation is large (d1 < 2c − a). Such a choice may hinder the diffusion of large innovations, which may be detrimental to society since large innovations are likely to be those bringing breakthroughs and opening big opportunities for technological improvements (cumulative innovation). Proposition 5 suggests one way to make innovators patent their very inventive innovations: this may be induced either by reducing the level of compulsory disclosure which is equivalent, in our model, to increasing the value of the parameter f or by increasing the value of the expected damages. Case 2: 2c − a < d1 <

9c−4a 5

(medium innovations)

In this case, the innovator is partially imitated under regime S (dS2 (d1 ) = 9c − 4a − 4d1 < d1 ) and its equilibrium expected profit under this regime is given by: ˜ S (d1 ) = ΠS (d1 , 9c − 4a − 4d1 ) = (3c − a − 2d1 )2 Π 1 1 18

Three subcases must be distinguished according to the value of dP2 (d1 , f, θ) which affects ˜ P (d1 , θ, f ) = ΠP (d1 , dP (d1 , f, θ)). Π 1

1

2

Subcase 2.1: f < M in(α (d1 , θ) , β (d1 , θ)) We know that for such values of parameters, the innovator is fully imitated under regime P (dP2 (d1 ) = d1 ). Its equilibrium expected profits under regime P are : 2 ˜ P (d1 , θ, f ) = ΠP (d1 , d1 ) = (a − d1 ) Π 1 1 (3 − θ)2

Some straightforward calculations lead to: ˜ 1 ) = 9c − 4a − 5d1 ˜ P (d1 , θ, f ) > Π ˜ S (d1 ) ⇐⇒ θ > θ(d Π 1 1 3c − a − 2d1

(8)

˜ 1 ) and to patent it if θ > Hence the innovator chooses to keep its innovation secret if θ < θ(d   ˜ 1 ). Note that θ(d ˜ 1 ) is a decreasing function of d1 ∈ 2c − a, 9c−4a such that θ( ˜ 9c−4a ) = 0 θ(d 5

5

˜ − a) = 1. and θ(2c ˜ 1 ) to the previously defined safe protection It is interesting to compare this new threshold θ(d ˜ 1 ) can be level θβ (d1 ). Since A(θ) is strictly decreasing, the comparison of θβ (d1 ) and θ(d ˜ 1 )). From β (d1 , θβ (d1 )) = 0 we derive: derived from the comparison of A(θβ (d1 )) and A(θ(d (d1 −2c+a)2 ˜ 1 ), one obtains: A(θ(d ˜ 1 )) = A(θβ (d1 )) = and, using the above expression of θ(d 2 9(a−d1 ) 3(d1 −2c+a)(3c−a−2d1 ) and 3 (3c − a − 2d1 ) . For . Therefore we just need to compare d1 −2c+a 9 (a−d1 )2  d1 −2c−a 9c−4a a−c any d1 ∈ 2c − a, 5 ,we have < 45 and 3 (3c − a − 2d1 ) > 3(a − c) and so we 9

˜ 1 )) which is equivalent to θ(d ˜ 1 ) < θβ (d1 ). get A(θβ (d1 )) < A(θ(d This result shows that even if a medium innovation is expected to be fully imitated under the patent regime, a patent protection is still preferred by the innovator if the patent holder expects to recover the infringer’s profit with a sufficiently high probability. Note that this ˜ 1 ) is lower than the safe protection level θβ (d1 ) previously defined. Therefore, probability θ(d patents will be filed even if their protection level is strictly lower than the safe protection level warranting perfect protection against imitation (see figure 4). Subcase 2.2 : α (d1 , θ) < f < γ(θ) In this subcase, the innovator is partially imitated under regime P (dP2 (d1 ) = dint 2 (d1 , f, θ)) and under regime S (dS2 (d1 ) = 9c − 4(a + d1 ) < d1 ). The following lemma (proven in appendix A3) compares these imitation levels under regime P and regime S. Lemma 6 When the innovator is partially imitated under both protection regimes, two cases arise: - If f < 9A(θ) then the innovator is more imitated under regime P than under regime S. - If f > 9A(θ) then the innovator is more imitated under regime S than under regime P. 19

Finally, by combining the two previous lemmas, we reach the conclusion that when f > 9A(θ), the innovator chooses to patent its innovation since the damage effect and the competition effect go in the same direction. However, if f < 9A(θ) the damage effect and the competition ˜ P (d1 , θ, f ) to Π ˜ S (d1 ) effect are opposite. The following lemma is crucial in order to compare Π 1

1

in this case.

Lemma 7 Along any curve f = KA(θ) in the (θ, f ) space, where K is a strictly positive ˜ P (d1 , θ, f ) increases in the patent strength θ parameter, the innovator’s equilibrium profit Π 1

as long as partial imitation occurs. Proof. We showed in appendix A3 that the follower’s level of imitation dint 2 (d1 , f, θ) depends on the parameters θ and f only through

f A(θ) .

This implies that dint 2 (d1 , f, θ) remains constant

as one moves on curve f = KA(θ). Then lemma 7 appears as a simple corollary of the result according to which the function θ −→ ΠP1 (d1 , d2 , θ) is increasing, for given marginal costs d1 and d2 , This result appeared when we introduced the damage effect. Using this lemma, we derive the following result (proven in appendix A4). Lemma 8 For medium innovations (2c − a < d1 <

9c−4a 5 ),

when α (d1 , θ) < f < 9A(θ),

there exists a threshold λ(d1 , θ) decreasing in the patent strength θ such that the innovator ˜ 1 ) and patents it if f > λ(θ, d1 ) or keeps its innovation secret if f < λ(θ, d1 ) and θ < θ(d ˜ 1 ).The threshlod function λ (d1 , θ) satisfies the following two conditions : λ(d1 , 0) = 1 θ > θ(d ˜ 1 )) = α(d1 , θ(d ˜ 1 )). and λ(d1 , θ(d The first condition states that the innovator is indifferent between patenting and keeping secrecy when θ = 0 and f = 1 and the second that it is indifferent between these two regimes ˜ 1 ) and f = α(d1 , θ(d ˜ 1 ) which is consistent with our previous findings (see figure when θ = θ(d 4). Subcase 2.3: β (d1 , θ) < f < α (d1 , θ) or f > M ax(α (d1 , θ) , γ(θ)) In this subcase, the innovator is not imitated at all under the patent regime (dP2 (d1 ) = d1 ). We derive from lemma 4 that the innovator’s optimal protection regime is regime (P). Finally, the following proposition summarizes the case 2c − a < d1 <

9c−4a 5 .


Proposition 9 For medium process innovations 2c − a < d1 < 9c−4a , there exist a thresh5 ˜ 1 ) decreasing in the innovation size c − d1 and a threshold function λ (d1 , θ) old function θ(d decreasing in the patent strength θ such that : ˜ 1 ) and f < λ (d1 , θ) then the innovator chooses the secrecy regime - If θ < θ(d ˜ 1 ) or f > λ (d1 , θ) then the innovator chooses the patent regime. - If θ > θ(d

20

This proposition, illustrated in figure 4, can be interpreted as follows. When the patent is ˜ 1 )) the innovator always chooses to patent its innovation whatever the strong enough (θ > θ(d imitation cost parameter f. This means that the imitation cost allowed by disclosure does ˜ 1 ). In particular, it patents its not matter anymore when the patent strength is above θ(d innovation even if disclosure makes the imitation costless. This result does not hold for weak ˜ 1 )). In this case, the competition effect and the damage effect may go in patents (θ < θ(d opposite directions leading to different protection regimes according to whether the disclosure effect of patenting is high enough (f < λ (d1 , θ)) or not. The effect of the innovation size on ˜ 1 ) decreases as the innovation size decreases (d1 increases), the secrecy region is clear: since θ(d the corresponding area shrinks as the innovation is smaller. Case 3: d1 >

9c−4a 5

(small innovations)

In this case, the innovator will be fully imitated if it chooses regime S (dS2 (d1 ) = d1 ). Therefore, according to lemma 4, the innovator’s optimal protection regime is the patent regime (P). 21

Proposition 10 Small innovations (d1 >

9c−4a 5 )

are always patented.

This result can be explained in the following way : since small innovations are fully imitated under secrecy, patenting is preferred for two reasons. First, it may deter imitation leading to higher market profits. Second, even when the patent strength and the disclosure effect are such that imitation cannot be deterred, it allows the innovator to expect some damages compensating its market profit loss due to imitation, which is not the case under secrecy regime. We develop now an overall discussion of our results by comparing them to those of Anton and Yao (2004) and embedding them in a broader perspective. The latter proposition states that inventors of small process innovations always prefer protection induced by patent to the protection conferred by a trade secret. This result is a reminder of the ”little patents” expression coined by Anton and Yao (2004). However, the argument behind this common result is different in our model. Under the secrecy regime, small process innovations are fully imitated for two reasons that reinforce one another. First, imitating a small innovation is not very costly and second, there is no threat of an infringement lawsuit when the innovation is kept secret. Under patent protection, such a threat exists and it may overrun the benefits that the infringer expects from imitating the leader. Note that in our model, small innovations may be imitated under the patent regime while this does not occur in Anton and Yao. For large process innovations, our results are similar to the ” big secrets” characterization obtained in Anton and Yao (2004). Large process innovations are never imitated when they are kept secret, while the enabling knowledge disclosed by a patent may reduce the imitation cost in a way that renders their imitation attractive. This classical tradeoff in the economics of patents explains why ”big secrets” are preferred to ”big patents”. Note however that our model does not totally discard the possibility of patenting some large process innovations. This may occur when their imitation under the patent regime is too costly. In this case we have shown that the innovator is indifferent between secret and patent, because in both cases, the innovation is not imitated. Finally, it is for medium process innovations that our results significantly differ from those of Anton and Yao. We have shown that keeping a medium process innovation secret does not avoid imitation. It is precisely for medium process innovations that partial innovation occurs under the secrecy regime. However, under the patent regime, imitation may be either absent, partial and total. We have also shown that the innovator may patent or keep secrecy while in Anton and Yao, medium process innovations are always patented and partially disclosed.

22

6

Licensing agreements as an alternative to litigation

In this section, we allow for licensing agreements between the innovator and the follower. Since our purpose is not to study all the possible agreements that may emerge, we restrict our attention to the simple case of process innovations leading to a small cost reduction, i.e. d1 >

9c−4a 5 .

We know from the previous section that those innovations are always patented.

We analyze licensing agreements between the innovator and an imitator that avoid litigation to be completed until the court’s decision. We study two-part tariff licenses (r, F ) where r is a royalty rate and F a fixed fee. Let us first examine the equilibrium outcomes when the innovator and the follower agree on a license (r, F ). Gross profits can be written as ΠL 1 (x1 , x2 , d1 , r, F ) = (a − x1 − x2 − d1 ) x1 + rx2 + F ΠL 1 (x1 , x2 , d1 , r, F ) = (a − x1 − x2 − (d1 + r)) x2 − F An equilibrium of the competition stage under the license regime (L) is given by xL 1 =

a − d1 + r 3

xL 2 =

a − d1 − 2r 3

and leads to the following equilibrium price: pL = We assume hereafter that 0 ≤ r ≤ since any royalty rate such that r

a + 2d1 + r 3

a−d1 2 . This assumption does not entail any loss of generality 1 1 > a−d leads to the same boundary solution as r = a−d 2 2 :

xL 1 =

a − d1 2

xL 2 =0 Equilibrium gross profits are given by: ˜L Π 1 (d1 , r, F ) =



a − d1 + r 3

˜L Π 2 (d1 , r, F ) =



2 +r

a − d1 − 2r +F 3

a − d1 − 2r 3

23

(9)

2 −F

(10)

6.1

A benchmark: the non-collusive agreement

First, we address the following question: does there exist a license (r, F ) such that the profits under the license regime replicate the profits under the patent regime (without licensing), i.e: ˜L ˜P Π 1 (d1 , r, F ) = Π1 (d1 , θ) and ˜ L (d1 , r, F ) = Π ˜ P (d1 , θ) Π 2 2 Since we do not set any restriction on the fixed fee F (in particular, we allow F to be negative), it is clear that such a question amounts to the existence of a royalty rate r such that the ˜ L (d1 , r, F ) + Π ˜ L (d1 , r, F ) under the license regime replicate the ˜ L (d1 , r) = Π industry profits Π 1 2 industry profits under the patent regime without licensing, i.e: ˜ L (d1 , r) = Π ˜ P1 (d1 , θ) + Π ˜ P2 (d1 , θ) Π

(11)

1 2−θ [−r2 + (a − d1 )r + 2(a − d1 )2 ] = (a − d1 )2 9 (3 − θ)2

(12)

which can be rewritten as:

Solving this equation, we find a unique solution : r˜(θ) =

θ (a − d1 ) 3−θ

(13)

The equilibrium price is then given by: p˜L (θ) =

a + d1 (2 − θ) 3−θ

The royalty rate r˜(θ) and equilibrium price p˜L (θ) are increasing in θ. We can now derive the ˜ L (d1 , r˜(θ), F˜ (θ)) = Π ˜ P (d1 , θ): fixed fee F˜ (θ) from the equation Π 2

2

θ(θ − 1) F˜ (θ) = (a − d1 )2 (3 − θ)2 Thus, we obtain the following properties of the function F˜ : - F˜ (0) = F˜ (1) = 0. - F˜ (θ) < 0 for any θ ∈ ]0, 1[ .     - F˜ (θ) is strictly decreasing over 0, 35 and strictly increasing over 53 , 1 . Note that the license (˜ r(θ), F˜ (θ)) not only replicates the expected profits of the innovator and the follower under regime P (without licensing) but also induces the same equilibrium 24

outputs and then the same equilibrium price. It is indeed easy to check that : x ˜L 1 (θ) =

a − d1 + r˜(θ) (1 − θ) (a − d1 ) = = xP1 (θ) 3 3−θ

and x ˜L 2 (θ) =

a − d1 − 2˜ r(θ) a − d1 = = xP2 (θ) 3 3−θ

which obviously lead to p˜L (θ) = pP (θ) where pP (θ) is the equilibrium price under the patent regime without licensing. Thus, the fact that competition stage outcomes are identical under both regimes (patent under the shadow of infringement and license defined by (˜ r(θ), F˜ (θ))), allows us to consider the license regime defined by (˜ r(θ), F˜ (θ)) as a benchmark to which we could compare the outcomes of the licensing agreements that are likely to emerge. Consider Shapiro’s criterion that a settlement, e.g. a licensing agreement, should not be allowed unless it does not make consumers worse off relative to the situation where litigation occurs. The non collusive agreement we fully characterized above meets this requirement since it does not result in a higher price. However, there exists a fundamuntal difference between Shapiro(2003) and our setting: competition occurs and prices are set in Shapiro’s model after the issues of validity and infringement are resolved while in our setting, it happens before (competition occurs under the shadow of infringement). Due to this difference, it is possible to design a settlement that improves firm’s profits without decreasing consumers’ expected surplus in Shapiro (2003) while this is not possible in our setting.7

6.2

The most collusive agreement

One of the agreements which are more likely to emerge is a two-part tariff license thet maximizes the firms’ joint profits. The part of the industry profits allocated to each firm is then determined by the fixed fee F .   Consider such a licensing agreement, denoted by rˆ(θ), Fˆ (θ) . Since rˆ(θ) is defined by rˆ(θ) = Arg max ΠL (d1 , r), the royalty rˆ(θ) does not depend on the patent strength θ. Moreover, it is 0≤r≤

a−d1 2

h i 1 easy to see that ΠL (d1 , r) is an increasing function of r over interval 0, a−d . This yields: 2 rˆ(θ) = rˆ =

a − d1 = r˜(1), ∀θ ∈ [0, 1] 2

7

In our model, the only agreement that is acceptable by both parties and does not harm consumers is the non collusive agreement determined above.

25

leading to the price: pˆ(θ) = pˆ =

a + d1 = p˜L (1) ≥ p˜L (θ) , ∀θ ∈ [0, 1] 2

This result means that the patent strength is no longer reflected by the price paid by consumers. In particular, low quality patents which generate lower prices when litigated, generate the same price as would high quality patents. This obviously harm the consumers. a−d1 −2r L Note that = 0 when r = rˆ. Hence, when the follower accepts the license 3   x2 = rˆ, Fˆ (θ) it implicitly accepts to stay out of the market and the industry profits are then

captured by the patentee. Nevertheless, the innovator transfers a part of these monopoly   profits to the licensee through the negative fixed fee Fˆ (θ). In other words, the license rˆ, Fˆ (θ) is equivalent to an agreement whereby the innovator pays its competitor to stay out of the market. Hence this illustrates in our framework the issue, raised by Shapiro (2003), that licensing agreements used to settle patent litigation could actually be used as a way to reach anticompetitive outcomes.   The patentee’s and the licensee’s profits when they agree on the license rˆ, Fˆ (θ) are given by: ΠL 1 (d1 , θ) =

(a − d1 )2 + Fˆ (θ) 4

ˆ ΠL 2 (d1 , θ) = −F (θ) This type of agreement is accepted by both the innovator and i the follower as long as the fixed h (a−d1 )2 P P ˆ fee F (θ) is in the interval Π (d1 , θ) − , −Π (d1 , θ) , that is:8 1

2

4

h i Fˆ (θ) ∈ −A(θ) (5 − θ) (a − d1 )2 , −A(θ) (a − d1 )2

(14)

The possibility that such two-part tariffs involving negative fees emerge in licensing agreements is a big concern for competition authorities. We know that in the pharmaceutical industry, agreements of this kind have been actually used by some patent holders in their negociations with generic challengers under the Hatch-Waxman Act. They obviously harm consumers and this is why patent settlements, which take the form of licensing agreements, must be under the scrutiny of competition authorities (Shapiro, 2003). 8 The license (ˆ r, F0 (θ)) where F0 (θ) = −A(θ) (5 − θ) (a − d1 )2 is the optimal license, from the innovator’s perspective, among all the licenses that maximize industry profits. It is likely to emerge if the innovator has a ”take it or leave it” bargaining power. Indeed, it is clear that with such a bargaining power the innovator will pay its competitor the minimum amount that makes it accept to stay out the market.

26

7

Conclusion

Departing from the usual convention that patents are perfect forms of protection opens a lot of research avenues. One of the most important issues is to know under what conditions patent protection is preferred to secrecy. Our model provides a theoretical answer to this question for a process innovation. For each class of cost reduction (small, medium and large) we have obtained specific results. First, we have determined the imitation level in each regime. Second, in the space of the two key parameters (patent strength and relative imitation cost) we have derived the partition that delineates areas where one protection regime dominates the other. How can one use these results for a policy purpose? This is an interesting and complex issue for which we suggest preliminary insights. Consider the relative imitation cost. In a world where patent design is independent of the invention, particularly concerning the same compulsory disclosure for all patents, it seems very hard to determine a priori what would be the value of the imitation cost parameter. One can simply reach a rather vague idea of the secrecy effectiveness of the invention, that leads to an idiosyncratic characterization covering a large spectrum of possibilities, running from the ”naked idea” case to the ”perfectly hidden idea” case. This type of assessment would depend on some priors on whether the invention could be more or less easily discovered by reverse-engineering. But in a world where a patent is not designed around the ”one size fits all” principle, some flexibility could be introduced by allowing each innovator to choose a patent inside a menu of characteristics. For instance an innovator may have to choose between a patent with strong property rights and high disclosure requirements and a patent with weak property rights and low disclosure requirements. If an incentive mechanism built around this principle could be achieved, it would be an appropriate answer to the rather disappointing result that ”little patents and big secrets” are the preferred forms of protection. Small innovations could be easily imitated because their rights are weak. Large innovations could be patented because their rights are strong. In both cases, diffusion of innovation would be enhanced. The construction of such an incentive mechanism is left for future research. Our model analyzes also the licensing agreements between a patent holder and a competitor. Such agreements avoid the litigation to go until completion. One of the possible consequences of a patent settlement as an alternative to a trial raises some concerns. The royalty rate paid by the licensee does not depend on the patent strength as a natural benchmark would command. Licensing very bad quality patents may occur with as high royalty rate as if the patent were full-proof. Moreover, the patentee pays a fixed fee to the licensee to compensate its loss in the market. While the two parties maximize their joint profits, it is clear that such a settlement harms consumers and creates a big concern for society. Shapiro (2003) and Farrell and Shapiro (2005) reach the same conclusion by using quite different models.

27

Finally, while some economists (Ayres and Klemperer, 1999) find that probabilistic rights open welfare improving opportunities (entry occurs under the shadow of punishment) it is also important to stress some of their negative consequences. Adopting trade secrecy for large inventions may reduce the diffusion of innovation. Moreover, patent settlements of the sort examined in this paper are detrimental to society. This is one reason why patent quality is probably one of the most challenging issues to which we are now confronted.

8

References

Anton, J.J. and D. A. Yao, 2004, ”Little patents and big secrets: managing intellectual property”, Rand Journal of Economics 35, 1-22. Anton, J.J. and D. A. Yao, 2005, ”Finding “Lost Profits”: an equilibrium analysis of patent infringement damages”, W.P. Duke University. Anton, J.J., H. Greene and D. A. Yao, 2005, ”Policy implications of weak patent rights”, mimeo. Bebchuk, L.A., ”Litigation and settlement under imperfect information”, Rand Journal of Economics, 15, 404-415. Chang, H. F. 1995. “Patent Scope, Antitrust Policy, and Cumulative Innovation,” Rand Journal of Economics 26, 34-57. Cohen, W.M, R.R. Nelson and J.P. Walsh, 2000, ”Protecting their intellectual assets: appropriability conditions and why US manufacturing firms patent or not”, NBER W.P. 7552. Crampes, C, 1986, ”Les inconv´enients d’un d´epˆot de brevet pour une entreprise innovatrice”, L’Actualit´e Economique, Revue d’Analyse Economique, 62, 4, 521-534. Crampes, C. and C. Langinier, Litigation and settlement in patent infringement cases,Rand Journal of Economics, 33(2), 258-274. Encaoua, D. and A. Hollander, 2004, ”Competition policy and innovation”, Oxford Review of Economic Policy 18, 63-79. Encaoua, D., D. Guellec and C. Martinez, 2005, ”Patent systems for encouraging innovation: Lessons from economic analysis”, W.P., University Paris I. Farrell, J. and C. Shapiro, 2005, ”How strong are weak patents?”, W. P. University of California at Berkeley. Gallini, N.T., 1992, ”Patent policy and costly innovation”, Rand Journal of Economics 23, 52-63. Green, J. and S. Scotchmer, 1995, ”On the division of profit in sequential innovation”, Rand Journal of Economics 26, 20-23. Horstman, I., G.M. MacDonald and A. Slivinski, 1985, ”Patents as information transfer mechanisms: To patent or (Maybe) not to patent”, Journal of Political Economy 95-5, 837-858.

28

Lemley, M.A. and C. Shapiro, 2005, ”Probabilistic patents”, Journal of Economic Perspectives 19-2, 75-98. Levin, R.C., A.K. Klevorick, R.R. Nelson and S.G. Winter, 1987, ”Appropriating the returns from industrial R&D”, Brooking Papers on Economic Activity, 783-820. Mansfield, E., 1986, ”Patents and innovation: an empirical survey”, Management Science 32, 173-181. Merges, R., 1999, ”As many as six impossible patents before breakfast: property rights for business concepts and patent system reform”, Berkeley High Technology Law Journal 14, 577-615. Meurer, M., 1989, ”The settlement of patent litigation”, Rand Journal of Economics, 20, 77-91. Pakes, A. and Z. Griliches, ”Patents and R&D at the firm level: a first look”, Economics Letters 5-4, 377-381. Scherer, F.M., 1965, ”Market structure, opportunity and the output of patented inventions”, American Economic Review 55, 1097-1125. Scherer, F.M., 1967, ”Market structure and the employment of scientists and engineers”, American Economic Review 57, 524-531. Scherer, F.M., 1983, ”The propensity to patent”, International Journal of Industrial Organization 1, 107-128. Shapiro, C., 2003, ”Antitrust analysis of patent settlements between rivals”, Rand Journal of Economics 34-2, 391-411. Scotchmer, S., 2004, Innovation and Incentives, The MIT Press, Cambridge, Ma. Scotchmer, S. and J. Green, 1990, ”Novelty and disclosure in patent law”, Rand Journal of Economics 21, 131-146.

9

Appendix

A1. Proof of Proposition 1 Since ΠP2 (d1 , d2 , f, θ) = 0 for any d2 ∈

h

i

a+d1 2 ,c

, it follows from the fact that imitation is

costly that the follower’s best imitation level over this interval is d2 = c : Arg maxiGP2 (d1 , d2 , f, θ) = c h

d2 ∈

(15)

a+d1 ,c 2

which implies that the follower’s optimal imitation level is necessarily equal to either c or Arg max i GP2 (d1 , d2 , f, θ). h

d2 ∈ d1 ,

a+d1 2

29

h i 1 In order to determine the maximum value of GP2 (d1 , d2 , f, θ) over the interval d1 , a+d , we 2 must distinguish two cases: Case 1 : f < 8A(θ) h i 1 The function d2 → GP2 (d1 , d2 , f, θ) is convex over d1 , a+d since GP2 (d1 , d2 , f, θ) = H(d1 , d2 , f, θ) 2 over this interval. Moreover, it is straightforward to show that dint 2 (d1 , f, θ) < dint 2 (d1 , f, θ)

Then,

dint 2 (d1 , f, θ)

> d1 or ≤ d1 .   P int > d1 then d2 → Gh2 (d1 , d2 , f, θ) is decreasing over the interval d , d (d , f, θ) 1 1 2 i a+d1 increasing over the interval dint , which entails that d2 → GP2 (d1 , d2 , f, θ) 2 (d1 , f, θ), 2

there are two possibilities according to whether - If

a+d1 2 .

dint 2 (d1 , f, θ)

and is

h i 1 reaches its maximum value over the interval d1 , a+d at either d2 = d1 or d2 = 2

a+d1 2 .

i h P (d , d , f, θ) is increasing over the interval d , a+d1 - If dint (d , f, θ) < d then d → G 1 1 2 1 2 1 2 2 2 which implies that it reaches its maximum value at d2 n= c. o 1 The crucial point is that in both cases, dP2 (d1 , f, θ) ∈ d1 , a+d , c . Since we know that the 2 follower prefers not to imitate rather than imitate at a level d2 = compare GP2 (d1 , d1 , f, θ) to GP2 (d1 , c, f, θ) = 0 in order to determine

a+d1 2 , it is sufficient to P d2 (d1 , f, θ). Hence, two

subcases arise:  2 1 - If f < 2A(θ) a−d then GP2 (d1 , d1 , f, θ) > 0 which results in dP2 (d1 , f, θ) = d1 (full c−d1 imitation)  2 1 - If 2A(θ) a−d < f < 8A(θ) then GP2 (d1 , d1 , f, θ) < 0 which results in dP2 (d1 , f, θ) = c (no c−d1 imitation). Case 2 : f > 8A(θ) a+d1 and d2 → GP2 (d1 , d2 , f, θ) is concave over the In dint 2 (d1 , f, θ) > c > 2 h this case, i h interval i a+d1 1 d1 , 2 . Then, the function d2 → GP2 (d1 , d2 , f, θ) is increasing over the interval d1 , a+d , 2

which results in Arg max i GP2 (d1 , d2 , f, θ) = h d2 ∈

a+d1 2 ,

a+d d1 , 2 1

a+d1 2 .

Since GP2 (d1 , d2 , f, θ) = 0 for any d2 ≥

we conclude that dP2 (d1 , f, θ) = c (no imitation).

A2. Proof of Proposition 2 The imitator must compare the maximal net profit it can get when it imitates, i.e. sup GP2 (d1 , d2 , f, θ) = sup H(d1 , d2 , f, θ) d2 ∈[d1 ,c[ (d1 −2c−a)2 .Two cases 9

, to the net profit it derives from keeping its old technology,

d2 ∈[d1 ,c[ ie. GP2 (d1 , c, f, θ)

must be distinguished:

Case 1 : f < 8A(θ) P In this case, dint 2 (d1 , f, θ) > c and d2 → G2 (d1 , d2 , f, θ) is convex over the interval [d1 , c[,

which entails that d2 → GP2 (d1 , d2 , f, θ) is decreasing over the interval [d1 , c[ , and results in sup GP2 (d1 , d2 , f, θ) = GP2 (d1 , d1 , f, θ) which has to be compared to GP2 (d1 , c, f, θ) . This

d2 ∈[d1 ,c[

leads us to distinguish two subcases. 30

=

Define β (d1 , θ) = 2A(θ)



a−d1 c−d1

2

−

2 9

2 d1 −2c+a . c−d1 > GP2 (d1 , c, f, θ)



- If f < β (d1 , θ) then GP2 (d1 , d1 , f, θ)

which results in dP2 (d1 , f, θ) = d1 (full

imitation). - If f > β (d1 , θ) then GP2 (d1 , d1 , f, θ) < GP2 (d1 , c, f, θ) which results in dP2 (d1 , f, θ) = c (no imitation). Case 2 : f > 8A(θ) P In this case, dint 2 (d1 , f, θ) < c and d2 → G2 (d1 , d2 , f, θ) is concave over the interval [d1 , c[.

Two subcases must be distinguished : P - If dint 2 (d1 , f, θ) < d1 then d2 → G2 (d1 , d2 , f, θ) is decreasing over the interval [d1 , c[ , which

implies that

sup GP2 (d1 , d2 , f, θ) = GP2 (d1 , d1 , f, θ).

d2 ∈[d1 ,c[

P - If dint 2 (d1 , f, θ) > d1 then d2 → G2 (d1 , d2 , f, θ) reaches its maximum over [d1 , c[ at d2 =

dint 2 (d1 , f, θ):

sup GP2 (d1 , d2 , f, θ) = GP2 (d1 , dint 2 (d1 , f, θ), f, θ).

d2 ∈[d1 ,c[

Consider the condition dint 2 (d1 , f, θ) < d1 . It is straightforward to show that this inequality can be rewritten as : f < α (d1 , θ) =

4(a − d1 ) A(θ) c − d1

Then, the two previous subcases can be written as: - If 8A(θ) < f <

4(a−d1 ) c−d1 A(θ)

then

sup GP2 (d1 , d2 , f, θ) = GP2 (d1 , d1 , f, θ) which has to be

d2 ∈[d1 ,c[

compared to GP2 (d1 , c, f, θ) (this has been previously done). - If f >

4(a−d1 ) c−d1 A(θ)

then

sup GP2 (d1 , d2 , f, θ) = GP2 (d1 , dint 2 (d1 , f, θ), f, θ) which has to be

d2 ∈[d1 ,c[

compared to GP2 (d1 , c, f, θ). Comparing these two terms is equivalent to comparing f to the threshold γ(θ) = • If

8 (3−θ)2 −9 1−θ

4(a−d1 ) c−d1 A(θ)

. More specifically:

P < f < γ(θ) then GP2 (d1 , dint 2 , f, θ) > G2 (d1 , c, f, θ) which leads to

dP2 (d1 , f, θ) = dint 2 (d1 , f, θ) (partial imitation). P P • If f > γ(θ) then GP2 (d1 , dint 2 , f, θ) < G2 (d1 , c, f, θ) which leads to d2 (d1 , f, θ) = c (no

imitation). Let us now show that the equations α (d1 , θ) = β (d1 , θ) and α (d1 , θ) = γ(θ) have the same solution θ0 (d1 ) over the interval [0, 1[ ,which means that the curves f = α (d1 , θ) , f = β (d1 , θ) and f = γ(θ) meet at the same point. Some straightforward computations show that the equation α (d1 , θ) = β (d1 , θ) is equivalent to the equation: A(θ) =

1 d1 − 2c + a 9 a − d1

31

Therefore, the equations α (d1 , θ) = β (d1 , θ) and α (d1 , θ) = γ(θ) have the same solution 1 d1 −2c+a 9 a−d1

in θ over the interval [0, 1[ if (and only if) 4(a−d1 ) c−d1 X

=

8 X−9 .

is a solution in X to the equation

It is easy to check that this is satisfied.

A3. Proof of lemma 6 It is easy to see that dint 2 (d1 , f, θ) depends on the parameters (f, θ) only through the parameter 2

f int int f (3−θ) 1−θ .With a slight modification of notations, we can write d2 (d1 , f, θ) = d2 (d1 , A(θ) ). It

is also clear that the imitation level dint 2 is increasing in from

dS2 (d1 )

=

f dint 2 (d1 , A(θ)

f A(θ) .

Hence, lemma 6 simply derives

= 9).

A4. Proof of lemma 8. Consider first the case θ < θ˜ (d1 ) . Let us show that equation ΠP1 (d1 , dint 2 (d1 , f, θ), θ) = ΠS1 (d1 , 9c − 4a − 4d1 ) which expresses that the innovator is indifferent between patenting and keeping secrecy (in this subcase) has a unique solution in f over the interval [α(d1 , θ), 9A(θ)]. Since the function f −→ dint 2 (d1 , f, θ) is strictly increasing over the interval [α(d1 , θ), 9A(θ)], this is equivalent to state that equation ΠP1 (d1 , d2 , θ) = ΠS1 (d1 , 9c − 4a − 4d1 ) has a unique   int solution in d2 over the interval dint 2 (d1 , α(d1 , θ), θ), d2 (d1 , 9A(θ), θ) . The latter interval can simply be written as [d1 , 9c − 4a − 4d1 ] .Note that the function Fθ : d2 −→ ΠP1 (d1 , d2 , θ) − ΠS1 (d1 , 9c − 4a − 4d1 ) is a convex parabolic function then it is either i/ increasing over [d1 , 9c − 4a − 4d1 ] or ii/ U-shaped over [d1 , 9c − 4a − 4d1 ] . Since θ < θ˜ (d1 ), we have Fθ (d1 ) < 0 (see subcase 2.1). We also know from lemma 4 that F (9c − 4a − 4d1 ) ≥ 0. It follows that in both cases i/ and ii/ equation Fθ (d2 ) = 0 has a unique solution over [d1 , 9c − 4a − 4d1 ] . S Hence, the equation ΠP1 (d1 , dint 2 (d1 , f, θ), θ) = Π1 (d1 , 9c − 4a − 4d1 ) has a unique solution in S f that we denote by λ(d1 , θ). Note that dint 2 (d1 , 1, 0) = d2 (d1 ) = 9c − 4a − 4d1 which leads

to λ(d1 , 1) = 0. Note also that : S ΠP1 (d1 , dint 2 (d1 , f, θ), θ) > Π1 (d1 , 9c − 4a − 4d1 ) if and only if f > λ (d1 , θ)

(16)

P S ˜ ˜ Furthermore, we know that dint 2 (d1 , α(d1 , θ (d1 )), θ (d1 )) = d1 and Π1 (d1 , d1 , θ) = Π1 (d1 , 9c − 4a−4d1 ) (see subcase 2.1) so ΠP (d1 , dint (d1 , α(d1 , θ˜ (d1 )), θ˜ (d1 )), θ˜ (d1 )) = ΠS (d1 , 9c−4a−4d1 ) 1

2

1

˜ 1 )) = α(d1 , θ(d ˜ 1 )). which leads to λ(d1 , θ(d Consider now the case θ > θ˜ (d1 ) . Let θ0 > θ˜ (d1 ) and f0 ∈ ]α(d1 , θ), 9A(θ)[ . The point (θ0 , f0 ) belongs to the curve f =

f0 A(θ0 ) A (θ) .

It is easy to see (graphically or analytically) ˜ 1 ) and that the curve f = necessarily meets either the curve defined by θ = θ(d f ≤ α(d1 , θ) or the curve defined by f = λ(d1 , θ) and θ ≤ θ˜ (d1 ) at a point (θ1 , f1 ) such that f0 A(θ0 ) A (θ)

θ1 < θ0 . Since in any point of the last two curves the innovator’s profit under regime P is equal to its profit under regime S, and θ1 < θ0 , we derive from lemma 7 that the innovator’s profit under regime P is greater than its profit under regime S when (θ, f ) = (θ0 , f0 ) . Hence, 32

the innovator chooses to patent its innovation. Let us now show that λ(d1 , θ) is strictly decreasing in θ.Consider θ1 and θ2 such that ˜ 1 ). The points (θ1 , d1 ) and θ2 , λ(d1 ,θ1 ) A (θ2 ) belong to the curve defined θ1 < θ2 ≤ θ(d A(θ1 ) λ(d1 ,θ1 ) λ(d1 ,θ1 ) P int A(θ1 ) A (θ). We derive from lemma 7 that Π1 (d1 , d2 (d1 , A(θ1 ) A (θ2 ) , θ2 ), θ2 ) S ΠP1 (d1 , dint 2 (d1 , λ (d1 , θ1 ) , θ1 ), θ1 ) = Π1 (d1 , 9c − 4a − 4d1 ) which leads, according to (17), λ(d1 ,θ1 ) A(θ1 ) A (θ2 ) > λ(d1 , θ2 ). Furthermore we know that A (θ) is positive and decreasing, A(θ2 ) A(θ1 ) < 1. This allows us to state that λ(d1 , θ1 ) > λ(d1 , θ2 ).

by f =

33

< to so